> [!definition] > > Let $X, Y$ be $E$ and $F$ [[Manifold|manifolds]] of class $C^p$ ($p \ge 1$). [[Split Subspace|Split]] $F = F_1 \oplus F_2$ and $W$ be a $F_1$-[[Submanifold|submanifold]] of $Y$. A [[Manifold Morphism|morphism]] $f: X \to Y$ is **transversal** over $W$ if for each $x \in f^{-1}(W)$, the composite map > $ > \begin{CD} > T_xX @>{T_xf}>> T_{f(x)}Y @>{\text{canon}}>> T_{f(x)}Y/T_{f(x)}W > \end{CD} > $ > is surjective and its kernel splits. In other words, the image of the [[Tangent Space|tangent space]] by [$df_x$](Differential.md) and $T_{f(x)}W$ span $T_{f(x)}Y$. > [!definition] > > Let $E$ be a [[Banach Space|Banach space]], then the diagonal > $ > \Delta = \bracs{(x, x): x \in E} > $ > is [[Closed Set|closed]] in $E^2$ and splits with $E \times 0$ or $0 \times E$. Therefore the diagonal is a closed submanifold. > > Let $X, Y, Z$ be Banach $C^p$-manifolds ($p \ge 1$) with $Z$ modelled on $E$, and $f: X \to Z$ and $g: Y \to Z$ be $C^p$-[[Manifold Morphism|morphisms]]. $f$ and $g$ are **transversal** if the morphism > $ > f \times g: X \times Y \to Z^2 > $ > is transversal over the diagonal.